How is it that a random sample gives a fairly accurate representation of public opinion?

Answer 1

It's obviously not completely accurate. But the more people you include the more accurate it gets. Statisticians choose the sample size to be large enough for a fairly accurate representation.

Answer 2

to ensure that every person in the population has an equal chance

of being picked for the poll

It is not practical to poll everybody, so a relatively small sample is taken. The sample may have random aspects, but a well-designed poll will try to include all relevant populations in a representative number.